Rina,

Heare are my thoughts,

1) there is no inconsistancy in saying (5 - x)^2 = (X - 5)^2 it is perfectly ok .. to be more clear assume a value of X say 3, you will have 5 - x = 5 - 3 = 2 and x - 5 = 3 - 5 = -2; now squar both the things ou will get (2^2) = ((-2)^2) = 4

2) -x * |x| > 0 means that x < 0 because we have |x| which is always positive or zero. Now that means - x has to be positive in order to satisfy the given condition. so x is negative i.e. x < 0

for the given question X < 0 < 5 means that x - 5 < 0

now |X - 5| = - (x - 5) .. if (x-5 < 0 ... (from definition of mode)

so |x - 5 | = 5 - x .. thus statement (2) alone is also sufficinet to answer the question.

Hope it clarifies.